Abstract
Let
(\((i = \sqrt { - 1;} t,x\)-real variables). It is proved that in the rectangle\(D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}\), the function h satisfies the followingfunctional inequality:
where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result
The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle.
Variation and complexity of solutions of Schrödinger equation are discussed.
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Communicated by Andrew M. Odlyzko
Acknowledgements and Notes. The author was supported by DEPSCOR Grant N000149611003 and NSF Grant No. DMS 9706883. The author expresses his gratitude to Irina Mitrea, who read the manuscript and made a number of valuable remarks.
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Oskolkov, K. Schrödinger equation and oscillatory Hilbert transforms of second degree. The Journal of Fourier Analysis and Applications 4, 341–356 (1998). https://doi.org/10.1007/BF02476032
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DOI: https://doi.org/10.1007/BF02476032